If I had a power series given by $\sum_{n=0}^{\infty}a_{n}x^{n}$ and it was the solution to a differential equation, how would I go about finding the radius of convergence of said power series? TIA
2026-04-12 02:00:33.1775959233
Radius of convergence of a power series solution to a differential equation
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Assuming this is a linear DE and it can be written
$$y^{(n)} + p_{n-1}(x)y^{(n-1)} + \cdot +p_0(x)y = g(x)$$,
then you can find all the singularities in all the coefficient functions $p_k(t)$. The distance from $0$ to the nearest singularity is the radius of convergence.
Notes: If you center the series at $x_0$, then it's the distance from $x_0$ to the nearest singularity.
You have to account for the complex singularities. If one of your coefficients is $1/(1+x^2)$ then the radius of convergence is no more than the distance from $0$ to $i$.